1. Field of the Invention
The invention relates in general to characterization of frequency dependent behavior of a passive electrical network and more particularly, the use of a Smith Chart to evaluate frequency dependent behavior at lower frequencies.
2. Description of the Related Art
The behavior of electrical networks consisting of passive linear devices such as, connectors, cables, micro strip lines and printed circuit boards may be frequency dependent. Characterization involves measurement of a network's response to excitation by signals at different frequencies. One measure of the frequency response of a network is its complex coefficient, which can be defined in terms of a ratio between a reflected voltage wave and an incident voltage wave. Typically, a network will manifest different coefficient values in response to excitation by signals at different frequencies.
FIG. 1 is an illustrative drawing showing an incident voltage Vinc and reflected voltage Vrefl in a network comprising a transmission line having characteristic impedance Z0 and a load having load impedance ZL. Impedance of a component or circuit is a measure of the component's or circuit's opposition to a sinusoidal alternating electric source. The coefficient (called Gamma and symbolized by F) is defined as,ΓL=Vrefl/Vinc=ZL−Z0/ZL+Z0=Γr+jΓi.
The amount of reflected signal from a load to the source is dependent upon the degree of mismatch between characteristic impedance and load impedance. Matching of characteristic impedance and load impedance avoids reflection of energy back from the load to the source. Specifically, to obtain the maximum power transfer from a source to a load, the characteristic impedance should be equal to the complex conjugate of the load impedance. In higher frequency environments such as video lines, RF and microwave networks, for example, spurious elements like wire inductances, interlayer capacitances and conductor resistances can have a significant yet unpredictable impact upon impedance matching. As a result, theoretical calculations and simulations may be insufficient to predict frequency dependent device behavior. Accordingly, actual measurements of device behavior at different excitation frequencies are obtained to characterize frequency dependency.
Characterization of passive devices or networks of passive devices over a range of different frequencies results in a frequency domain signature for the devices or the network of devices. Signal processing techniques can be used to predict network behavior in the time domain from a device frequency signature based on the frequency domain signature. For instance, the inverse Fourier transform often is used to obtain the time-domain response of circuits and systems from measured frequency-domain data. In order for the Fourier transform to function properly, however, information throughout the entire spectrum should be provided. Theoretically, this spectrum spans from DC (zero frequency) to infinity. In reality, since the Fourier process is discretized and the data is truncated, the highest frequency information is considered as infinite frequency data. In general, the inverse fast Fourier transform (IFFT) algorithm expects to receive a frequency domain signature that comprises a list of data points corresponding to uniformly spaced frequency points the first of which will be considered as DC and the last of which will be treated as infinite frequency. Therefore, the Fourier transform algorithm ordinarily treats the first measured data point as the DC (zero frequency) information. If the first data point in the list (which corresponds to the lowest measured frequency) is not an actual DC point, then there can be serious inaccuracies in the inverse transform process.
Unfortunately, in many cases, the lowest frequency at which measured data points are available is much higher than DC. Current state-of-the-art equipment measurement equipment typically can provide network parameter data at frequencies no lower than about 45 MHz. For instance, data from systems characterized with a standard HP8510C/110 GHz Network Analyzer ordinarily do not contain information for frequencies below 45 MHz. For RF and Microwave applications, this is in general not a problem. For signal integrity analysis on digital PCB/Packaging, however, it can be a serious problem since the digital spectrum is broadband, from DC to daylight, and the DC value of the network parameter is critical to the correctness of entire transient waveforms in time domain.
In the past, extrapolation techniques have been used to compensate for missing low frequency signal measurement information. However, these prior methods generally assume that certain data points are relatively close to DC. The correctness of the extrapolated data obtained through these algorithms depends at least in part upon the assumption that certain data points are close to DC, and in many cases that assumption may not be accurate.
For example, one prior approach is to use a measurement value obtained for the lowest measured frequency as if it was the DC frequency value. Unfortunately this can be an inaccurate approximation of the DC value. For instance, the accuracy of DC measurement data may be diminished if the lowest frequency is beyond the onset of the skin effect region. Another prior approach is to extrapolate the magnitude and phase of a network parameter separately. Still another approach is to extrapolate the real and imaginary parts of the parameter separately. These latter two approaches assume that the DC value can be linearly extrapolated from the last two measured data values. However, the validity of this assumption is case dependent. It depends on the specific structures (e.g., simple uniform transmission line or 3D discontinuity structures) and the range of missing frequency data at low frequency end.
Thus, there has been a need for improvement in the acquisition of information concerning low frequency behavior of a network. In particular, there has been a need for improvement in the acquisition of information concerning the behavior of networks at frequencies below the frequency at which behavior ordinarily can be measured using typical measurement equipment. The present invention meets these needs.